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Theorem moabs 2541
Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2567. (Revised by BJ, 14-Oct-2022.)
Assertion
Ref Expression
moabs (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Proof of Theorem moabs
StepHypRef Expression
1 ax-1 6 . 2 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑))
2 nexmo 2539 . . 3 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
3 id 22 . . 3 (∃*𝑥𝜑 → ∃*𝑥𝜑)
42, 3ja 186 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑)
51, 4impbii 209 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1776  ∃*wmo 2536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965
This theorem depends on definitions:  df-bi 207  df-ex 1777  df-mo 2538
This theorem is referenced by:  mo3  2562  mo4  2564  moeu  2581  dffun7  6595  wl-mo3t  37557
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